# Find the standard deviation of a log-normal AR1-process with exponent

I am trying to replicate a RBC model with technology shock

$\log(z_{t+1})=\rho \log(z_t)+\epsilon_{t+1} \$ with $\epsilon_t \sim$ i.i.d.$\mathcal{N}(0,\sigma^2)$ and $0 < \rho < 1$

The original source states that:

$\sigma$ is determined so that the innovation in $z^{1-\theta}$ has standard deviation 0.007

where $0 < \theta < 1$ Unfortunately, I don't know how to approach this problem with a (non-integer) exponent and what exactly is meant by the innovation in $z^{1-\theta}$

How can I (analytically) find the value of $\sigma$?

Thanks for your help!

P.S. The source is Greenwood, Rogerson, Wright (1995), "Household Production in Real Business Cycle Theory", in Frontiers of Business Cycle Research, p.167